The barcodes produced by persistent homology and persistent (relative) cohomology are the same [1]}; a majority of existing state of the art solvers therefore employ variants of the cohomology algorithm.
Existing applied and theoretical work seeks to leverage this asymmetry to accelerate computation of generators in homology.
The Eirene library accomplishes this via a right-looking block reduction method which iteratively computes Schur complements and is therefore (in principle) agnostic to row versus column operations. The Ripserer library accomplishes this by first identifying pivot elements via elementary row operations, thereby reducing the scope of column operations to pivot columns [2]}; this offers impressive computational advantages, as non-pivot columns account for a disproportionate number of algebraic operations.
Recently, a purely row-based algorithm to compute generators in PH, developed via categorical homotopy theory, has been proposed in [3]}.
These methods relate, in a loose conceptual sense, to several other acceleration techniques which leverage knowledge about order, sparsity, and linear dependence to reduce the number of algebraic operations needed to perform persistence computations [4]}, [5]}, [6]}, [7]}, [8]}, [9]}.
